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The strong predictable representation property of semi-martingales and the notion of enlargement of filtration meet naturally in modeling financial markets, and theoretical problems arise. Here, first, we illustrate some of them through classical examples. Then, we review recent results obtained by studying predictable martingale representations for filtrations enlarged by means of a full process, possibly with accessible components in its jump times. The emphasis is on the non-uniqueness of the martingale enjoying the strong predictable representation property with respect to the same enlarged filtration. [ABSTRACT FROM AUTHOR]

Let and be an m-dimensional -semi-martingale and an n-dimensional -semi-martingale, respectively, on the same probability space , both enjoying the predictable representation property. We propose two representation results for the square-integrable -martingales, where . As a first application, we identify the biggest possible value of the multiplicity in the sense of Davis and Varaiya of , where, fixed , is the reference filtration of a martingale , which enjoys the -predictable representation property. This result helps us to identify a basis of martingales for the Poisson filtration enlarged by a general random time. A second application falls into the framework of credit risk modelling and in particular into the study of progressive enlargement of the market filtration by a default time. We present a new proof of the analogous of classical Kusuoka's theorem, when the risky asset price is a multidimensional semi-martingale enjoying the predictable representation property and the default time satisfies the density hypothesis. [ABSTRACT FROM AUTHOR]

We present two examples of loss of the predictable representation property for semi-martingales by enlargement of the reference filtration. First of all we show that the predictable representation property for a semi-martingaleXdoes not transfer from the reference filtrationto a larger filtrationif the information starts growing up to a positive time. Then we study the casewhen there exists a second special semi-martingaleYenjoying the predictable representation property with respect to. We establish conditions under which the triplet (X, Y, [X, Y]) enjoys the predictable representation property with respect to. [ABSTRACT FROM AUTHOR]

In this paper we study the effect of rare mutations, driven by a marked point process, on the evolutionary behavior of a population. We derive a Kolmogorov equation describing the expected values of the different frequencies and prove some rigorous analytical results about their behavior. Finally, in a simple case of two different quasispecies, we are able to prove that the rarity of mutations increases the survival opportunity of the low fitness species. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study approximation schemes for a nonlinear filtering problem of a partially observed diffusive system when the state process is the solution of a stochastic delay diffusion equation with a constant time lag and the observation process is a noisy function of the state. The approximating state is the linear interpolation of a modified Milstein scheme, which is asymptotically optimal with respect to the mean square -error within the class of all pathwise approximations based on equidistant observations of the driving Brownian motion. Upper bounds for the error of the filter approximations are computed. Some other discretization schemes for the state process are also considered. [Copyright &y& Elsevier]

We study an approximation scheme for a nonlinear filtering problem when the state process X is the solution of a stochastic delay diffusion equation and the observation process is a noisy function of X(s) for s ϵ [t - τ, t], where τ is a constant. The approximating state is the piecewise linear Euler-Maruyama scheme, and the observation process is a noisy function of the approximating state. The rate of convergence of this scheme is computed. [ABSTRACT FROM AUTHOR]

The aim of this paper is to give some approximation results for a class of nonlinear filtering problems with delay in the observation. First, we point out some general results on the approximation problem for the filter in nonlinear filtering. In particular, we give a general procedure for obtaining some upper bounds for the different approximations we consider. This procedure is then applied in the case of nonlinear filtering problems with delay (X, Y), which can be represented by means of a Markov system (X, Y∧), in the sense that Yt = Y∧a(t). Finally, these upper bounds are computed explicitly in the particular case of Markov jump process with counting observations. [ABSTRACT FROM AUTHOR]